Consistency Of Maximum Likelihood Estimator Research Articles

The Topp-Leone-Gompertz-Exponentiated Half Logistic-G Family of Distributions with Applications

This paper introduces and investigates a new family of distributions called the Topp-Leone-Gompertz-exponentiated half logistic-G (TL-Gom-EHL-G) distribution. Some mathematical and statistical properties of this family of distributions are derived. To estimate and evaluate the model parameters, the maximum likelihood estimation technique is used, and the consistency of maximum likelihood estimators is examined using Monte Carlo simulation. Applications to three real data sets from different areas were used to demonstrates the usefulness and versatility of the TL-Gom-EHL-G family of distributions.

Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications

ABSTRACT In this work, a new family of distributions referred to as type II exponentiated half-logistic-Gompertz-G (TIIEHL-Gom-G) family of distributions is introduced and studied. Some of the main statistical properties of these family of distributions are derived. The model parameters are estimated using the maximum likelihood estimation technique and consistency of maximum likelihood estimators is evaluated by performing a simulation study. The importance and versatility of the TIIEHL-Gom-G family of distributions is demonstrated in an application to two real data sets from different fields.

Identifiability of Asymmetric Circular and Cylindrical Distributions

Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sine-skewed circular distributions, including the sine-skewed von Mises and sine-skewed wrapped Cauchy distributions, and that of a M\"{o}bius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sine-skewed wrapped Cauchy distribution, and conditional distributions of a random variable on the non-negative real line given the circular random variable are a Weibull distribution and a generalized Pareto-type distribution, respectively.

Asymptotic Posterior Normality of Multivariate Latent Traits in an IRT Model

The asymptotic posterior normality (APN) of the latent variable vector in an item response theory (IRT) model is a crucial argument in IRT modeling approaches. In case of a single latent trait and under general assumptions, Chang and Stout (Psychometrika, 58(1):37–52, 1993) proved the APN for a broad class of latent trait models for binary items. Under the same setup, they also showed the consistency of the latent trait’s maximum likelihood estimator (MLE). Since then, several modeling approaches have been developed that consider multivariate latent traits and assume their APN, a conjecture which has not been proved so far. We fill this theoretical gap by extending the results of Chang and Stout for multivariate latent traits. Further, we discuss the existence and consistency of MLEs, maximum a-posteriori and expected a-posteriori estimators for the latent traits under the same broad class of latent trait models.

Mleand Bayes Estimator of Utilization Factor in Power Supply Queuing Model

In this paper,we have been obtainedMLE and Bayes Estimator of utilization factor in Power Supplyqueuing model based on requirement of power forthe number of customers at various sampled time points. We have been checked the consistency of maximum likelihood estimator of utilization factor and find the confidence interval for expected MLE of it.Bayes estimators of utilization factor havealso been obtained.We also derive the expression for Expectation of MLE of ρ, Bayes estimate of ρ under squared error loss function. Fordifferent choices of hyper-parametersthe minimum posterior risk of this Bayes estimator has also been obtained numerically by using R software packages.

On the Consistency of Maximum Likelihood Estimators for Causal Network Identification

We consider the problem of identifying parameters of a particular class of Markov chains, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph. Incoming edges to a node in the graph indicate that the state of the node at a particular time instant is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular node's state variable is a convex combination of the parental node states in the previous time instant and an additional Bernoulli noise random variable. This letter focuses on the problem of edge weight identification using Maximum Likelihood (ML) estimation and proves that the ML estimator is strongly consistent for two variants of the BAR model. We additionally derive closed-form estimators for the aforementioned two variants and prove their strong consistency.

Consistent maximum likelihood estimation using subsets with applications to multivariate mixed models

We present new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. Our theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full data. It requires neither that the data consist of independent observations, nor that the observations can be modeled as a stationary stochastic process. Compared to existing asymptotic theory using the idea of subsets, we substantially weaken the assumptions, bringing them closer to what suffices in classical settings. We apply our theory in two multivariate mixed models for which it was unknown whether maximum likelihood estimators are consistent. The models we consider have nonstochastic predictors and multivariate responses which are possibly mixed-type (some discrete and some continuous).

Weibull Inverse Lomax Distribution

A four-parameter lifetime model, named the Weibull inverse Lomax (WIL) is presented and studied. Some structural properties are derived. The estimation of the model parameters is performed based on Type II censored sample. Maximum likelihood estimators along with asymptotic confidence intervals of population parameters and reliability function are constructed. The property of consistency of maximum likelihood estimators has been verified on the basis of simulated samples. Â Further, the results are applied on two real data.

On scale Fisher consistency of maximum likelihood estimator for the exponential regression model under arbitrary frailty

It is demonstrated that the mle for the exponential regression model is Fisher consistent up to a scale factor under arbitrary frailty, a large class of cumulated hazards and normal regressors. Comparative simulations show good properties of the estimator.

On consistency of the MLE under finite mixtures of location-scale distributions with a structural parameter

We provide a general and rigorous proof for the strong consistency of maximum likelihood estimators of the cumulative distribution function of the mixing distribution and structural parameter under finite mixtures of location-scale distributions with a structural parameter. The consistency results do not require the parameter space of location and scale to be compact. We illustrate the results by applying them to finite mixtures of location-scale distributions with the component density function being one of the commonly used density functions: normal, logistic, extreme-value, or t. An extension of the strong consistency results to finite mixtures of multivariate elliptical distributions is also discussed.

Strong Consistency of Maximum Likelihood Estimators in Sequential-Cumulative Logit Model

In this article, for the sequential-cumulative logit model, we show that maximum likelihood estimates of regression parameter vector is asymptotically existence and strongly consistent under mild conditions

Consistency of maximum likelihood estimation for some dynamical systems

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Our proof involves ideas from both information theory and dynamical systems. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures.

Consistency of maximum likelihood estimation for some dynamical systems

Consistency of maximum likelihood estimation for some dynamical systems

CONSISTENCY UNDER SAMPLING OF EXPONENTIAL RANDOM GRAPH MODELS.

The growing availability of network data and of scientific interest in distributed systems has led to the rapid development of statistical models of network structure. Typically, however, these are models for the entire network, while the data consists only of a sampled sub-network. Parameters for the whole network, which is what is of interest, are estimated by applying the model to the sub-network. This assumes that the model is consistent under sampling, or, in terms of the theory of stochastic processes, that it defines a projective family. Focusing on the popular class of exponential random graph models (ERGMs), we show that this apparently trivial condition is in fact violated by many popular and scientifically appealing models, and that satisfying it drastically limits ERGM's expressive power. These results are actually special cases of more general results about exponential families of dependent random variables, which we also prove. Using such results, we offer easily checked conditions for the consistency of maximum likelihood estimation in ERGMs, and discuss some possible constructive responses.

Consistency of maximum likelihood estimators in a large class of deconvolution models

AbstractWe consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada

Consistency of maximum likelihood estimators for the regime-switching GARCH model

The regime-switching GARCH (generalized autoregressive conditionally heteroscedastic) model incorporates the idea of Markov switching into the more restrictive GARCH model, which significantly extends the GARCH model. However, the statistical inference for such an extended model is rather difficult because observations at any time point then depend on the whole regime path and the likelihood becomes intractable quickly as the length of observations increases. In this paper, by transforming it into an infinite order ARCH model, we obtain the possibility of writing a likelihood which can be handled directly and the consistency of the maximum likelihood estimators is proved. Simulation studies to illustrate the consistency and asymptotic normality of the estimators (for both Gaussian and non-Gaussian innovations) and a model specification problem are presented.

Strong consistency of maximum likelihood estimators for a discrete-time random field HJM-type interest rate model

We consider a discrete time Heath–Jarrow–Morton-type forward interest rate model, where the interest rate curves are driven by a geometric spatial autoregression field. Strong consistency of maximum likelihood estimators is proved for stable and unstable no-arbitrage models containing a simple stochastic discounting factor.

Estimation of the Rate–Distortion Function

Motivated by questions in lossy data compression and by theoretical considerations, we examine the problem of estimating the rate-distortion function of an unknown (not necessarily discrete-valued) source from empirical data. Our focus is the behavior of the so-called "plug-in" estimator, which is simply the rate-distortion function of the empirical distribution of the observed data. Sufficient conditions are given for its consistency, and examples are provided to demonstrate that in certain cases it fails to converge to the true rate-distortion function. The analysis of its performance is complicated by the fact that the rate-distortion function is not continuous in the source distribution; the underlying mathematical problem is closely related to the classical problem of establishing the consistency of maximum likelihood estimators. General consistency results are given for the plug-in estimator applied to a broad class of sources, including all stationary and ergodic ones. A more general class of estimation problems is also considered, arising in the context of lossy data compression when the allowed class of coding distributions is restricted; analogous results are developed for the plug-in estimator in that case. Finally, consistency theorems are formulated for modified (e.g., penalized) versions of the plug-in, and for estimating the optimal reproduction distribution.

Consistency and normality of estimates of hydrological extremes derived from a model for fragmented data

Summary This paper deals with a non-standard application of frequency analysis of extremes of hydrologic variables in which some years of record are incomplete. If F ( y ) is the cumulative distribution function (cdf) of the extreme variable in a complete year (or ‘block’), the cdf of the variable in an incomplete year is taken as F ( y ) p , where 0 p F ( y ) are estimated by maximum likelihood (ML), the usual large-sample characteristics of ML estimators (consistency, asymptotic Normality) may be modified. The paper examines the consistency, bias and approach to Normality of ML estimates of Gumbel parameters for position and scale, and of the Gumbel extreme event y 100 with 100-year return period. In the non-standard model F ( y ) p , consistency of ML estimates of Gumbel position and scale parameters is considerably modified, but the consistency of the estimated y 100 much less so. Estimates of y 100 are negatively biased, but the bias is similar to that found in the standard (no missing data) case. The results are relevant where hydrologic records are short and incomplete, such that all existing data must be fully utilized.

Statistical Inference of Adaptive Designs with Binary Responses

Adaptive designs of clinical trials are ethical alternatives when the traditional randomization becomes ethically infeasible in desperate medical situations. However, such a design creates a dependency among trial data and its statistical analysis becomes more complex than the analysis for traditional randomized clinical trials. In this article, we examine adaptive designs with dichotomous responses from two treatments and extend some commonly used statistical methods for independent data. Under a regularity condition, the estimated odds ratio and its logarithm are shown to follow asymptotically normal distributions. Moreover, the ordinary goodness-of-fit test statistic for two-by-two contingency tables with dependent data is shown to be asymptotically chi-square distributed. We also discuss the consistency of maximum likelihood estimators of the unknown parameters for a wide class of adaptive designs.